Earliest Known Uses of Some of the Words of Mathematics (V)

Last revision: Sept. 10, 2007

VANDERMONDE DETERMINANT, MATRIX etc. It seems that the determinant named after
Alexandre-Theophile
Vandermonde (1735-96) was never discussed by him.
Lebesgue believed that the attribution was due to a misreading of Vandermonde’s
unfamiliar notation. (DSB and H Lebesgue, "L'oeuvre mathématique de
Vandermonde," Enseignement Math. (2) 1 (1956),
203-223.). It is not known who
coined the term but it appears in 1888 in G. Weill, "Sur une forme du
déterminant de Vandermonde," Nouv. Ann. (1888).

The term VANISHING POINT was coined by Brook Taylor
(1685-1731), according to Franceschetti (p. 500).

The term VARIABLE was introduced by Gottfried Wilhelm Leibniz
(1646-1716) (Kline, page 340).

Variable is found in English as an adjective in 1710 in
Lexicon Technicum by J. Harris: "Variable Quantities,
in Fluxions, are such as are supposed to be continually increasing or
decreasing; and so do by the motion of their said Increase or
Decrease Generate Lines, Areas or Solidities" (OED2).

Variable is found in English as a noun in 1816 in a
translation of Lacroix’s Differential and Integral Calculus:
"The limit of the ratio..will be obtained by dividing the
differential of the function by that of the variable" (OED2).

VARIANCE, ANALYSIS OF VARIANCE, and ANOVA. The term variance was introduced by Ronald
Aylmer Fisher in 1918 in a paper on population genetics,
The Correlation Between Relatives on the Supposition
of Mendelian InheritanceTransactions of the Royal Society of
Edinburgh, 52, 399-433: "It is ... desirable in analysing the
causes of variability to deal with the square of the standard deviation as the
measure of variability. We shall term this quantity the Variance ..." (p. 399)

The phrase analysis of variance appears in the table
of contents of the same paper, but not in the text. It appears in the text of
a non-technical exposition Fisher gave, "The Causes of Human Variability,"
Eugenics Review, 10, (1918), 213-220. In these pieces the analysis
of variance refers to a decomposition of the population variance based on genetical
theory.

The later literature found two models in Fisher’s statistical
work and produced contrasting terms for them. C. Eisenhart in "The Assumptions
Underlying the Analysis of Variance," Biometrics, 3, (1947), 1-21
called them Model I and Model II analysis of variance. Eisenhart
also used the terms fixed effects and random effects, though more
informally. These appear with greater ceremony in H. Scheffé’s "Alternative
Models for the Analysis of Variance," Annals of Mathematical Statistics,
27, (1956), 251-271. Components of variance, another common term,
was introduced by H. E. Daniels in 1939 in his "The
Estimation of Components of Variance," Supplement to the Journal of
the Royal Statistical Society, 6, 186-197.

The acronym ANOVA
is used in a 1949 article by John W. Tukey: "Dyadic anova: an analysis
of variance for vectors" (Human Biology, 21, 65-110).

See also
ERROR, F-DISTRIBUTION, HYPOTHESIS and HYPOTHESIS-TESTING, STANDARD
DEVIATION, z-DISTRIBUTION.

[This entry was contributed by John Aldrich, based on OED2,
David (1995) and J. F. Box (1978) R. A. Fisher: The Life of a Scientist,
New York, Wiley.]

Pearson and some later statisticians, including R. A. Fisher, used the term
in the sense of the modern random variable: "The variable quantity,
such as the number of children, is called the variate, and the frequency
distribution specifies how frequently the variate takes each of its possible
values." Statistical Methods for Research Workers
(1925, p. 4.) (OED2)

When Gerhard Tintner (Variate Difference Method (1940))
suggested the use of differencing when calculating the variance of the error
around the trend of a single series the term correlation was no longer
appropriate and was dropped.

See also SPURIOUS CORRELATION.

VARIETY (as in modern algebraic geometry) was first used by E.
Beltrami in 1869 [Joseph Rotman].

Birkhoff used the term equationally defined
algebras in his AMS Colloquium Volume Lattice Theory in
the first 1940, second 1948 and third 1967 edition.

Hanna Neumann (1914-1971) introduced the term variety in "On
varieties of groups and their associated near-rings," Math.
Zeits., 65, 36-69 (1956) and popularised the term in her 1967
book Varieties of Groups [Phill Schultz].

VECTOR, VECTOR ANALYSIS and VECTOR SPACE. The entries below
follow changes in the use of the term vector over the past 150 years
or so. The story begins with a technical term in theoretical astronomy, "radius
vector," in which "vector" signified "carrier." In the first and biggest change
"radius" was dropped and "vector" was given a place in the algebra of quaternions
(c. 1840). The recognition that vectors are more interesting than quaternions,
especially in physics, led to vector analysis (c. 1880). In the 20th
century vector was extended from triples of numbers to n-tuples
and then to elements of abstract linear spaces.

The word VECTOR (which, like the word vehicle,
derives ultimately from the Latin vehĕre to carry) was first a technical
term in astronomical geometry. The OED’s earliest entry
is from a technical dictionary of 1704: J. Harris Lexicon Technicum I.
s.v., "A Line supposed to be drawn from any Planet moving round a Center, or
the Focus of an Ellipsis, to that Center or Focus, is by some Writers of the
New Astronomy, called the Vector; because 'tis that Line by which the Planet
seems to be carried round its Center."

Vector usually appeared in the phrase radius vector. The French term was rayon vecteur and can be found in e.g. Laplace’s
Traité de mécanique céleste
(1799-1825).

The term rayon vecteur is used in a non-astronomical context by Monge
"Application de l'Analyse à la Géométrie" (1807).
On p. 24 he writes, "on nomme la droite r le RAYON VECTEUR du point, et
l'origine des coordonnées devient un
pôle, d'où partent les rayons vecteurs des différens points de l'espace."
Monge does not seem to use the terminology later in the text, however. Cauchy in his
Leçons
sur les Applications du Calcul Infinitésimal à la Géométrie
of 1826 uses the term freely after introducing it in the initial
"Preliminaries" chapter: "Une droite AB, menée d'un point
A supposé fixe à un point B suppose mobile, sera
généralement désignée sous le nom de rayon vecteur." (p.14) (Information
from François Ziegler.) Radius vector appears n English in 1831 in Elements of the Integral
Calculus (1839) by J. R. Young: "...when the angle Ω
between the radius vector and fixed axis is taken for the independent variable,
the formula is...."

Hamilton would create a new meaning for vector (see VECTOR & SCALAR)
but he used radius vector in the conventional way in
On a General
Method in DynamicsPhilosophical
Transactions Royal Society (part II for 1834, pp. 247-308); see article 14.

A list of matrix and linear algebra terms having entries on this web site is
here.

Both terms appear in
"On quaternions"
a paper presented by Hamilton at a meeting of the Royal Irish Academy on November 11, 1844. This
paper adopts the convention of denoting a vector by a single (Greek) letter,
and concludes with a discussion of formulae for applying rotations to vectors
by conjugating with unit quaternions. It is on pages 1-16 in volume 3 of the
Proceedings of the Royal Irish Academy, covering the years 1844-1847,
and the volume is dated 1847. The following is from page 3:

On account of the facility with which this so called imaginary
expression, or square root of a negative quantity, is constructed by a right
line having direction in space, and having x, y, z for its three rectangular
axes, he has been induced to call the trinomial expression itself, as well as
the line which it represents, a VECTOR. A quaternion may thus be said to consist
generally of a real part and a vector. The fixing a special attention on this
last part, or element, of a quaternion, by giving it a special name, and denoting
it in many calculations by a single and special sign, appears to the author
to have been an improvement in his method of dealing with the subject: although
the general notion of treating the constituents of the imaginary part as coordinates
had occurred to him in his first researches.

The following is from page 8:

It is, however, a peculiarity of
the calculus of quaternions, at least as lately modified by the author, and
one which seems to him important, that it selects no one direction in space
as eminent above another, but treats them as all equally related to that extra-spacial,
or simply SCALAR direction, which has been recently called "Forward."

In Hamilton’s time radius-vector was an established
term in astronomy (see previous entry). Hamilton explains that he is giving
the term vector a new sense in Lecture I of his
Lectures
on Quaternions. In article 17 (p. 16) he described the difference between vector and radius-vector:

17. To illustrate more fully the
distinction which was just now briefly mentioned, between the meanings of the
"Vector" and the "Radius Vector" of a point, we may remark
that the RADIUS-VECTOR, in astronomy, and indeed in geometry also, is usually
understood to have only length; and therefore to be adequately expressed by
a SINGLE NUMBER, denoting the magnitude (or length) of the straight line which
is referred to by this usual name (radius-vector) as compared with the magnitude
of some standard line, which has been assumed as the unit of length. Thus, in
astronomy, the Geocentric Radius-Vector of the Sun is, in its mean value, nearly
equal to ninety-five millions of miles: if, then, a million of miles be assumed
as the standard or unit of length, the sun’s geocentric radius-vector is equal
(nearly) to, or is (approximately) expressible by, the number ninety-five: in
such a manner that this single number, 95, with the unit here supposed, is (at
certain seasons of the year) a full, complete and adequate representation or
expression for that known radius vector of the sun. For it is usually the sun
itself (or more fully the position of the sun’s centre) and NOT the Sun’s radius-vector,
which is regarded as possessing also certain other (polar) coordinates of its
own, namely, in general, some two angles, such as those which are called the
Sun’s geocentric right-ascension and declination; and which are merely associated
with the radius-vector, but not inherent therein, nor belonging thereto...

But in the new mode of speaking which
it is here proposed to introduce, and which is guarded from confusion with the
older mode by the omission of the word "RADIUS," the VECTOR of the
sun HAS (itself) DIRECTION, as well as length. It is, therefore NOT sufficiently
characterized by ANY SINGLE NUMBER, such as 95 (were this even otherwise rigorous);
but REQUIRES, for its COMPLETE NUMERICAL EXPRESSION, a SYSTEM OF THREE NUMBERS;
such as the usual and well-known rectangular or polar co-ordinates of the Sun
or other body or point whose place is to be examined...

A VECTOR is thus (as you will afterwards more clearly see)
a sort of NATURAL TRIPLET (suggested by Geometry): and accordingly we shall
find that QUATERNIONS offer an easy mode of symbolically representing every
vector by a TRINOMIAL FORM (ix + jy + kz); which form brings
the conception and expression of such a vector into the closest possible connexions
with Cartesian and rectangular co-coordinates.

Hamilton, in his "Lectures on Quaternions", is not
satisfied with having introduced vector. Within a few pages we find vectum,
vehend, revector, provector, provectum, transvehend, transvectum, etc.,
and identities such as

Provectum = Provector + Vector + Vehend.

Vector and scalar also appear in 1846 in a paper
"On Symbolical Geometry,"
in the The Cambridge and Dublin Mathematical Journal vol. I:

If then we give the name of scalars to all numbers
of the kind called usually real, because they are all contained on the one scale
of progression of number from negative to positive infinity [...]

Next Hamilton goes on to tell us about another "chief
class" of the "geometrical quotients," namely

the class in which the dividend is a line perpendicular to
the divisor. A quotient of this latter class we shall call a vector,
to mark its connection (which is closer than that of a scalar) with the conception
of space [...]

David Wilkins believes that the paper "On quaternions"
in the Proceedings of the Royal Irish Academy probably appeared earlier
than the CDMJ, probably some time in the first half of 1845.

The separation of the real and imaginary parts of a quaternion
is an operation of such frequent occurrence, and may be regarded as being so
fundamental in this theory, that it is convenient to introduce symbols which
shall denote concisely the two separate results of this operation. The algebraically
real part may receive, according to the question in which it occurs,
all values contained on the one scale of progression from number from
negative to positive infinity; we shall call it therefore the scalar part,
or simply the scalar of the quaternion, and shall form its symbol by
prefixing, to the symbol of the quaternion, the characteristic Scal., or simply
S., where no confusion seems likely to arise from using this last abbreviation.
On the other hand, the algebraically imaginary part, being geometrically
constructed by a straight line, or radius vector, which has, in general, for
each determined quaternion, a determined length and determined direction in
space, may be called the vector part, or simply the vector of
the quaternion; and may be denoted by prefixing the characteristic Vect. or
V...

Information for this article was provided by David Wilkins
and Julio González Cabillón. See HAMILTON and QUATERNION.

VECTOR ANALYSIS appears in 1881 in
J. W. Gibbs’s
(1839-1903) Elements of Vector Analysis Arranged for the Use of Students in Physics:
"The numerical description of a vector
requires three numbers, but nothing prevents us from using a single number for its symbolical
designation. An algebra or analytical method in which a single letter or other
expression is used to specify a vector may be called a vector algebra
or vector analysis." Scientific PapersII, (1906)
p. 17.

Prefacing the Elements is the statement,

"The fundamental principles of the following analysis
are such as are familiar under a slightly different form to students of quaternions.
The manner in which the subject is followed is somewhat different ..., since the
object of the writer does not require any use of the conception of the quaternion,
being simply to give a suitable notation for those relations between vectors,
or between vectors and scalars, which seem most important, and which lend themselves
most readily to analytical transformations, and to explain some of these transformations."

A prominent quaternionist, P. G. Tait, wrote, "Even Prof.
Willard Gibbs must be ranked as one of the retarders of Quaternion progress..."
After quoting this remark, M. J Crowe continues, "Gibbs did retard quaternion
progress for his ... Elements of Vector Analysis marks the beginning of
modern vector analysis." (A Historyof Vector Analysis (2nd
edition, p. 150)) Crowe considers
Oliver Heaviside
(1850-1925) an independent and nearly simultaneous creator of the modern system.

VECTOR FIELD is found in 1905 in “The Present Problems of Geometry”
by Dr. Edward Kasner in Congress of Arts and Science, Universal Exposition,
St. Louis, 1904: “The vector field deserves to be introduced as a standard form into geometry.”
[Google print search]

VECTOR PRODUCT and SCALAR PRODUCT are found in 1878 in
William Kingdon Clifford’s
(1845-1879) Elements ofDynamic (1878, p. 95): "We are led to two different kinds of product
of two vectors,..a vector product..and a scalar product." (OED2).

M. J Crowe A History of Vector Analysis (2nd
edition, pp. 139-43) finds an ambiguity in Clifford’s treatment of the scalar
product and argues that the modern definition only appears in the posthumously
published Common Sense of the Exact Sciences (1885), the relevant part
of which was written by the editor, Karl Pearson. Crowe presents Clifford as
a transitional figure in the movement from QUATERNION
algebra to VECTOR ANALYSIS.

VECTOR TRIPLE PRODUCT and SCALAR TRIPLE PRODUCT
appear in 1901 in E. B. Wilson, Vector Analysis: A Text Book
for the Use of Students of Mathematics and Physics Founded upon the Lectures
of J. Willard Gibbs: "The second triple product is the scalar product of
two vectors, of which one is itself a vector product, as A · (B
× C) or (A × B) · C. This sort of product has a
scalar value and consequently is often called the scalar triple product." (p.
68) and then on p. 71: "The third type of triple product is the vector product
of two vectors of which one is itself a vector product. Such are A ×
(B × C) and (A × B) × C.... This product is
termed the vector triple product in contrast to the scalar product." Both products
had appeared in Gibbs’s Elements of Vector Analysis but were not given
names.

VECTORS and VECTOR SPACE. At the beginning of the 20th century
vector usually meant a 3-dimensional object, though occasionally the term might be
applied to objects in 2 or n-dimensions. The situation is illustrated in the writing of
Maxime Bôcher.
In "The Theory of Linear Dependence" Bôcher wrote that "the simplest geometrical interpretation
for a complex quantity with n principal units is as a vector in space
of n dimensions." (Annals of Mathematics, 2, 1900/1, p.
83.) yet there are no vectors in his Introduction
to Higher Algebra (1907), where n-tuples
of numbers are called "complex quantities" or "sets of numbers."
It was years before the use of vector terminology became
standard in discussions of n-dimensional and abstract linear spaces.

The study of linear spaces began around the same time as Hamilton’s first work on quaternions with
Hermann Grassmann’s
Die Ausdehnungslehre (1844). In 1888
G. Peano
gave an abstract treatment in the 20th century manner in Calcolo geometrico secondo l'Ausdehnungslehre
di H. Grassmann preceduto dalle operazioni della logica deduttiva. See the St. Andrews page
Abstract linear spaces
for these developments.

[This entry was contributed by John Aldrich.
For more information see two articles in Historia Mathematica, 22,
(1995): J.-L. Dorier "A General Outline of the Genesis of Vector Space Theory"
(pp. 227-261) and G. H. Moore "The Axiomatization of Linear Algebra: 1875-1940"
(pp. 262-305).]

VENN DIAGRAM. The diagrams now known as Venn diagrams were introduced
in July 1880 in Venn, John: “On the
Diagrammatic and Mechanical Representation of Propositions and Reasonings,”
Philosophical Magazine and Journal of Science, X (1880), 1-18.

In chapter V of his Symbolic Logic (1881)
John Venn
showed a "reformed scheme of diagrammatic notation" based on what
has become known as the Venn diagram. According to Venn,
"the majority of modern logical treatises make at any
rate occasional appeal to diagrammatic aid ..."
but he considered the most popular
scheme, the "Eulerian," inapplicable for "the purposes of a really general Logic."

OED 2 traces the evolution of the term Venn diagram
through the following quotations: "The application of Mr Venn’s diagrammatic
scheme to syllogistic reasonings" in J. N. Keynes Studies & Exercises
in Formal Logic III. v. 207 (1884); "Syllogisms in Barbara and Camestres
may be taken in order to show how Dr Venn’s diagrams can be used." Ibid.
(ed. 3) III. iv. 298 (1894);
"This method resembles nothing so much as solution by means of the Venn diagrams."
C. I. Lewis Survey Symbolic Logic i. 77 (1918).

Information for this entry was contributed by John Aldrich and Wilfried Neumaier.
See also Euler diagram.

VERSED SINE. According to Smith (vol. 2, page 618), "This
function, already occasionally mentioned in speaking of the sine, is
first found in the Surya Siddhanta (c. 400) and, immediately
following that work, in the writings of Aryabhata, who computed a
table of these functions. A sine was called the jya; when it
was turned through 90 degrees and was still limited by the arc, it
became the turned (versed) sine, utkramajya or
utramadjya."

The Arabs spoke of the sahem, or arrow, and the word passed
over into Latin as sagitta.

Boyer (page 278) seems to imply that sinus versus appears in
1145 in the Latin translation by Robert of Chester of al Khowarizmi’s
Algebra, although Boyer is unclear.

In Practica geomitrae, Fibonacci used the term sinus versus
arcus. According to Smith (vol. 2), Fibonacci (1220) used
sagitta.

Fincke used the term sinus secundus for the versed sine.

Regiomontanus (1436-1476) used sinus versus for the versed
sine in De triangulis omnimodis (On triangles of all kinds;
Nuremberg, 1533).

Maurolico (1558) used sinus versus major (Smith vol. 2).

The OED shows a use in 1596 in English of "versed signe" by W.
Burrough in Variation of Compasse.

The term VERSIERA was coined by Luigi Guido Grandi (1671-1742)
(DSB). See witch of Agnesi.

The term VERSOR was introduced by William Rowan Hamilton (1805-1865). It appears in his Elements of Quaternions ii. i. (1866) 133:
“We shall now say that every Radial Quotient is a Versor. A Versor has
thus, in general, a plane, an axis, and an angle.” (OED and Julio González Cabillón.)
Unlike VECTORand SCALAR, two of Hamilton’s other QUATERNION terms, versor did not enter general currency.

VERTEX occurs in English in 1570 in John Dee’s preface to
Billingsley’s translation of Euclid (OED2).

In 1828, Elements of Geometry and Trigonometry (1832) by David
Brewster (a translation of Legendre) has:

In ordinary language, the word angle is
often employed to designate the point situated at the vertex. This
expression is inaccurate. It would be more correct and precise to
use a particular name, such as that of vertices for
designating the points at the corners of a polygon or of a polyedron.
The denomination vertices of a polyedron, as employed by us,
is to be understood in this sense.

VERTICAL ANGLE is found in English in 1571 in
A Geometrical Practical Treatize Named Pantometria edited by
Thomas Digges: "Two right lines crossing one another, make the contrary or verticall angles equall" [OED2].

The term appears in English translations of Euclid.

VIGINTIANGULAR. The OED2 shows one citation, from 1822, for
this term, meaning "having 20 angles." The word also appears in
Webster’s New International Dictionary, 2nd ed. (1934).

VINCULUM and VIRGULE. In the Middle Ages, the
horizontal bar placed over Roman numerals was called a
titulus. The term was used by Bernelinus. It was used more
commonly to distinguish numerals from words, rather than to indicate
multiplication by 1000.

Fibonacci used the Latin words virga and virgula for
the horizontal fraction bar.

In 1594 Blundevil in Exerc. (1636) referred to the fraction
bar as a "little line": "The Numerator is alwayes set above, and the
Denominator beneath, having a little line drawne betwixt them thus
1/2 which signifieth one second or one halfe" (OED2).

In 1660 J. Moore in Arith. used separatrix for the line
that was then placed after the units digit in decimals: "But the best
and most distinct way of distinguishing them is by a rectangular line
after the place of the unit, called Seperatrix. ... Therefore in
writing of decimall parts let the seperatrix be always used" (OED2).

In 1696, Samuel Jeake referred to the fraction bar as "the
intervening line" in his Arithmetick.

In 1771 separatrix was used for the fraction bar in Luckombe,
Hist. Printing: "The Separatrix, or rule between the Numerator
and Denominator [of fractions]" (OED2).

Leibniz, writing in Latin, used vinculum for the grouping
symbol.

In mathematics, vinculum originally referred only to the
grouping symbol, but some writers now use the word also to describe
the horizontal fraction bar.

VITAL STATISTICS. The term entered currency in the 1830s. The OED quotes
William Farr writing in
1837 in McCulloch’s Descriptive and
Statistical Account of the British Empire II. 567: “Vital Statistics;
or, the Statistics of Health, Sickness, Diseases, and Death.” Vital statistics
were one of the main interests of the statistical societies founded around that
time: including the London Statistical Society
and the American
Statistical Association. See STATISTICS

VON MISES DISTRIBUTION and VON MISES-FISHER
DISTRIBUTION. These terms are used for
directional distributions that are analogues of the normal distribution. The
distribution on the circle was introduced by
R. von Mises
in 1918 in connection with the hypothesis that atomic weights are
integers subject to error: "Ueber die 'Ganzahligkeit' der Atomgewicht und vervante
Fragen, Physikal. Z., 18, 490-500.

The modern interest in these distributions dates from the
early 1950s. E. J. Gumbel, J. A. Greenwood & D. Durand wrote
about "The Circular Normal Distribution: Theory and Tables," Journal of the
American Statistical Association, 48, (1953), 131-152. Greenwood
& Durand refer to this as the "Mises distribution law" in "The Distribution
of Length and Components of the Sum of n Random Unit Vectors," Annals
of Mathematical Statistics, 26, (1955), 233-246.

The distribution on the sphere was developed by
R. A. Fisher
in 1953 to treat problems in paleomagnetism:
Dispersion on a Sphere,
Proceedings of the Royal Society, A, 217, 295-305. Fisher appears not to have known
of von Mises’s work or, indeed, of Langevin’s derivation of the spherical distribution
in 1905 for a problem in statistical mechanics.

The name von Mises-Fisher distribution was
used by P. Hartman & G. S. Watson to refer to the n-dimensional generalisation
of the von Mises and Fisher distributions: "'Normal' Distribution
Functions on Spheres and the Modified Bessel Functions,"Annals of Probability, 2, (1974), 593-607

[This entry was contributed by John Aldrich, based on K. V. Mardia "Directional Distributions" Encyclopedia
of the Statistical Sciences, vol 2 (1982).]

The term VON NEUMANN ALGEBRAS was used by Jacques Dixmier in 1957 in Algebras of
operators in Hilbert space (von Neumann algebras). The term is named
for John
von Neumann, who had used the term ”rings of operators.” Another term is
”W-algebras.”

VULGAR FRACTION. In Latin, the term was fractiones
vulgares, and the term originally was used to distinguish
an ordinary fraction from a sexagesimal.

Trenchant (1566) used fraction vulgaire (Smith vol. 2,
page 219).

Digges (1572) wrote "the vulgare or common Fractions."

Sylvester used the term in On the theory of vulgar fractions,
Amer. J. Math. 3 (1880).